Is it possible to have a linear transformation that shifts a given matrix?

Question

Suppose I have a matrix A. I understand that I could for example rotate the vectors, shear, and flip them. But how could I move the entire matrix left or right?

Answer 1

Your problem is encountered in robotics, where rigid motions, i.e. rotations + translations can be expressed in just a matrix form, called "homogeneous transformations". Here, an example:

Given the affine mapping $f(x)=Ax+b$, define the followings homogeneous variable of $x$ as:

$x'=\begin{bmatrix} x\\1 \end{bmatrix}$

You can verify that, if $y=Ax+b$, then:

$y'=\begin{bmatrix} y\\1 \end{bmatrix}=\begin{bmatrix} A&b\\0&1 \end{bmatrix}\begin{bmatrix} x\\1 \end{bmatrix}=\begin{bmatrix} Ax+b\\1 \end{bmatrix}$.

Therefore, $M=\begin{bmatrix} A&b\\0&1 \end{bmatrix}$ is the matrix you are looking for. Be careful, since the homogeneous coordinates of a vector requires the dimension of your variable to be increased by 1. This algebraic trick has a beautiful geometric interpretation. Have a look on Projective Geometry if you are interested in it!